Scientific Machine Learning (SciML) has emerged as an active research field aiming to combine the strengths of computational modeling and data-driven methodologies. Integrating machine learning (ML) with physics-based partial differential equations [1] or thermodynamic principles [2] has proven effective for state prediction and for gaining insight into underlying physical mechanisms. However, the interpretability of learned models remains a major challenge. In many existing approaches, physical knowledge is either restricted to a small set of trainable parameters, embedded in black-box components, or encoded through data-driven invariants that lack clear physical meaning. Physically-Guided Neural Networks with Internal Variables have shown promise for model selection and for identifying nonlinear material behavior [3], but unlike classical parametric identification, they generally lack statistically grounded decision rules. To address this limitation, we introduce the Reliability-Aware Physically-Guided Neural Network with Internal Variables (RAPGNNIV) framework. This approach explicitly incorporates data uncertainty during training and propagates it to the model’s explanatory components, enabling uncertainty-aware identification of constitutive relationships. By prescribing a finite set of candidate models, RAPGNNIV supports statistically grounded model identification, plausibility assessment, and the construction of hyper-robust predictions [4]. We demonstrate that the framework can recover nonlinear constitutive equations from unstructured datasets under heterogeneous loading conditions, combining the expressive power of ML for equation discovery with interpretability and uncertainty awareness. The resulting models yield both accurate predictions under unseen conditions and interpretable representations of the underlying physical mechanisms.